udacity mpc -to-line

模型预测控制求解

2018-10-17_145518.jpg 2018-10-17_145459.jpg

模型

2018-10-17_150046.jpg
2018-10-17_150054.jpg

#include "MPC.h"
#include <math.h>
#include <cppad/cppad.hpp>
#include <cppad/ipopt/solve.hpp>
#include "Eigen-3.3/Eigen/Core"
#include "Eigen-3.3/Eigen/QR"
#include "matplotlibcpp.h"
#include <time.h>
namespace plt = matplotlibcpp;

using CppAD::AD;

// We set the number of timesteps to 25
// and the timestep evaluation frequency or evaluation
// period to 0.05.
size_t N = 10;//**预测步长**
double dt = 0.1;//**采样时间**

// This value assumes the model presented in the classroom is used.
//
// It was obtained by measuring the radius formed by running the vehicle in the
// simulator around in a circle with a constant steering angle and velocity on a
// flat terrain.
//
// Lf was tuned until the the radius formed by the simulating the model
// presented in the classroom matched the previous radius.
//
// This is the length from front to CoG that has a similar radius.
const double Lf = 2.67;

// Both the reference cross track and orientation errors are 0.
// The reference velocity is set to 40 mph.
double ref_v = 40;

// The solver takes all the state variables and actuator
// variables in a singular vector. Thus, we should to establish
// when one variable starts and another ends to make our lifes easier.**变量个数索引**
size_t x_start = 0;
size_t y_start = x_start + N;
size_t psi_start = y_start + N;
size_t v_start = psi_start + N;
size_t cte_start = v_start + N;
size_t epsi_start = cte_start + N;
size_t delta_start = epsi_start + N;
size_t a_start = delta_start + N - 1;

class FG_eval {
 public:
  Eigen::VectorXd coeffs;
  // Coefficients of the fitted polynomial.
  FG_eval(Eigen::VectorXd coeffs) { this->coeffs = coeffs; }

  typedef CPPAD_TESTVECTOR(AD<double>) ADvector;
  // `fg` is a vector containing the cost and constraints.
  // `vars` is a vector containing the variable values (state & actuators).
  void operator()(ADvector& fg, const ADvector& vars) {
    // The cost is stored is the first element of `fg`.
    // Any additions to the cost should be added to `fg[0]`.
    fg[0] = 0;

    // The part of the cost based on the reference state.**目标函数 误差部分**
    for (int t = 0; t < N; t++) {
      fg[0] += CppAD::pow(vars[cte_start + t], 2);
      fg[0] += CppAD::pow(vars[epsi_start + t], 2);
      fg[0] += CppAD::pow(vars[v_start + t] - ref_v, 2);
    }

    // Minimize the use of actuators.**目标函数 输入部分**
    for (int t = 0; t < N - 1; t++) {
      fg[0] += CppAD::pow(vars[delta_start + t], 2);
      fg[0] += CppAD::pow(vars[a_start + t], 2);
    }

    // Minimize the value gap between sequential actuations.**目标函数 输入跳动部分**
    for (int t = 0; t < N - 2; t++) {
      fg[0] += CppAD::pow(vars[delta_start + t + 1] - vars[delta_start + t], 2);
      fg[0] += CppAD::pow(vars[a_start + t + 1] - vars[a_start + t], 2);
    }

    //
    // Setup Constraints
    //
    // NOTE: In this section you'll setup the model constraints.**10个步长的等式约束**

    // Initial constraints
    //
    // We add 1 to each of the starting indices due to cost being located at
    // index 0 of `fg`.
    // This bumps up the position of all the other values.
    fg[1 + x_start] = vars[x_start];
    fg[1 + y_start] = vars[y_start];
    fg[1 + psi_start] = vars[psi_start];
    fg[1 + v_start] = vars[v_start];
    fg[1 + cte_start] = vars[cte_start];
    fg[1 + epsi_start] = vars[epsi_start];

    // The rest of the constraints
    for (int t = 1; t < N; t++) {
      // The state at time t+1 .
      AD<double> x1 = vars[x_start + t];
      AD<double> y1 = vars[y_start + t];
      AD<double> psi1 = vars[psi_start + t];
      AD<double> v1 = vars[v_start + t];
      AD<double> cte1 = vars[cte_start + t];
      AD<double> epsi1 = vars[epsi_start + t];

      // The state at time t.
      AD<double> x0 = vars[x_start + t - 1];
      AD<double> y0 = vars[y_start + t - 1];
      AD<double> psi0 = vars[psi_start + t - 1];
      AD<double> v0 = vars[v_start + t - 1];
      AD<double> cte0 = vars[cte_start + t - 1];
      AD<double> epsi0 = vars[epsi_start + t - 1];

      // Only consider the actuation at time t.
      AD<double> delta0 = vars[delta_start + t - 1];
      AD<double> a0 = vars[a_start + t - 1];

      AD<double> f0 = coeffs[0] + coeffs[1] * x0;
      AD<double> psides0 = CppAD::atan(coeffs[1]);

      // Here's `x` to get you started.
      // The idea here is to constraint this value to be 0.
      //
      // Recall the equations for the model:
      // x_[t+1] = x[t] + v[t] * cos(psi[t]) * dt
      // y_[t+1] = y[t] + v[t] * sin(psi[t]) * dt
      // psi_[t+1] = psi[t] + v[t] / Lf * delta[t] * dt
      // v_[t+1] = v[t] + a[t] * dt
      // cte[t+1] = f(x[t]) - y[t] + v[t] * sin(epsi[t]) * dt
      // epsi[t+1] = psi[t] - psides[t] + v[t] * delta[t] / Lf * dt
      fg[1 + x_start + t] = x1 - (x0 + v0 * CppAD::cos(psi0) * dt);
      fg[1 + y_start + t] = y1 - (y0 + v0 * CppAD::sin(psi0) * dt);
      fg[1 + psi_start + t] = psi1 - (psi0 + v0 * delta0 / Lf * dt);
      fg[1 + v_start + t] = v1 - (v0 + a0 * dt);
      fg[1 + cte_start + t] =
          cte1 - ((f0 - y0) + (v0 * CppAD::sin(epsi0) * dt));
      fg[1 + epsi_start + t] =
          epsi1 - ((psi0 - psides0) + v0 * delta0 / Lf * dt);
    }
  }
};

//
// MPC class definition
//

MPC::MPC() {}
MPC::~MPC() {}

vector<double> MPC::Solve(Eigen::VectorXd x0, Eigen::VectorXd coeffs) {
  size_t i;
  typedef CPPAD_TESTVECTOR(double) Dvector;

  double x = x0[0];
  double y = x0[1];
  double psi = x0[2];
  double v = x0[3];
  double cte = x0[4];
  double epsi = x0[5];


  // number of independent variables
  // N timesteps == N - 1 actuations**变量数量**
  size_t n_vars = N * 6 + (N - 1) * 2;
  // Number of constraints
  size_t n_constraints = N * 6;

  // Initial value of the independent variables.
  // Should be 0 except for the initial values.
  Dvector vars(n_vars);
  for (int i = 0; i < n_vars; i++) {
    vars[i] = 0.0;
  }
  // Set the initial variable values**初始化变量及变量上下限**
  vars[x_start] = x;
  vars[y_start] = y;
  vars[psi_start] = psi;
  vars[v_start] = v;
  vars[cte_start] = cte;
  vars[epsi_start] = epsi;

  // Lower and upper limits for x
  Dvector vars_lowerbound(n_vars);
  Dvector vars_upperbound(n_vars);

  // Set all non-actuators upper and lowerlimits
  // to the max negative and positive values.
  for (int i = 0; i < delta_start; i++) {
    vars_lowerbound[i] = -1.0e19;
    vars_upperbound[i] = 1.0e19;
  }

  // The upper and lower limits of delta are set to -25 and 25
  // degrees (values in radians).
  // NOTE: Feel free to change this to something else.
  for (int i = delta_start; i < a_start; i++) {
    vars_lowerbound[i] = -0.436332;
    vars_upperbound[i] = 0.436332;
  }

  // Acceleration/decceleration upper and lower limits.
  // NOTE: Feel free to change this to something else.
  for (int i = a_start; i < n_vars; i++) {
    vars_lowerbound[i] = -1.0;
    vars_upperbound[i] = 1.0;
  }

  // Lower and upper limits for constraints
  // All of these should be 0 except the initial
  // state indices.**都是等式约束，上下限都设置为0，初始值的约束**
  Dvector constraints_lowerbound(n_constraints);
  Dvector constraints_upperbound(n_constraints);
  for (int i = 0; i < n_constraints; i++) {
    constraints_lowerbound[i] = 0;
    constraints_upperbound[i] = 0;
  }
  constraints_lowerbound[x_start] = x;
  constraints_lowerbound[y_start] = y;
  constraints_lowerbound[psi_start] = psi;
  constraints_lowerbound[v_start] = v;
  constraints_lowerbound[cte_start] = cte;
  constraints_lowerbound[epsi_start] = epsi;

  constraints_upperbound[x_start] = x;
  constraints_upperbound[y_start] = y;
  constraints_upperbound[psi_start] = psi;
  constraints_upperbound[v_start] = v;
  constraints_upperbound[cte_start] = cte;
  constraints_upperbound[epsi_start] = epsi;

  // Object that computes objective and constraints
  FG_eval fg_eval(coeffs);

  // options
  std::string options;
  options += "Integer print_level  0\n";
  options += "Sparse  true        forward\n";
  options += "Sparse  true        reverse\n";

  // place to return solution
  CppAD::ipopt::solve_result<Dvector> solution;

  // solve the problem
  CppAD::ipopt::solve<Dvector, FG_eval>(
      options, vars, vars_lowerbound, vars_upperbound, constraints_lowerbound,
      constraints_upperbound, fg_eval, solution);

  //
  // Check some of the solution values
  //
  bool ok = true;
  ok &= solution.status == CppAD::ipopt::solve_result<Dvector>::success;

  auto cost = solution.obj_value;
  std::cout << "Cost " << cost << std::endl;
  return {solution.x[x_start + 1],   solution.x[y_start + 1],
          solution.x[psi_start + 1], solution.x[v_start + 1],
          solution.x[cte_start + 1], solution.x[epsi_start + 1],
          solution.x[delta_start],   solution.x[a_start]};
}

//
// Helper functions to fit and evaluate polynomials.
//

// Evaluate a polynomial.
double polyeval(Eigen::VectorXd coeffs, double x) {
  double result = 0.0;
  for (int i = 0; i < coeffs.size(); i++) {
    result += coeffs[i] * pow(x, i);
  }
  return result;
}

// Fit a polynomial.
// Adapted from
// https://github.com/JuliaMath/Polynomials.jl/blob/master/src/Polynomials.jl#L676-L716
Eigen::VectorXd polyfit(Eigen::VectorXd xvals, Eigen::VectorXd yvals,
                        int order) {
  assert(xvals.size() == yvals.size());
  assert(order >= 1 && order <= xvals.size() - 1);
  Eigen::MatrixXd A(xvals.size(), order + 1);

  for (int i = 0; i < xvals.size(); i++) {
    A(i, 0) = 1.0;
  }

  for (int j = 0; j < xvals.size(); j++) {
    for (int i = 0; i < order; i++) {
      A(j, i + 1) = A(j, i) * xvals(j);
    }
  }

  auto Q = A.householderQr();
  auto result = Q.solve(yvals);
  return result;
}

int main() {
  MPC mpc;
  int iters = 50; //**仿真时长**

  Eigen::VectorXd ptsx(2);
  Eigen::VectorXd ptsy(2);
  ptsx << -100, 100;
  ptsy << -1, -1;

  // The polynomial is fitted to a straight line so a polynomial with
  // order 1 is sufficient.
  auto coeffs = polyfit(ptsx, ptsy, 1);//**拟合跟踪轨迹**

  // NOTE: free feel to play around with these，**初始值**
  double x = -1;
  double y = 10;
  double psi = 0;
  double v = 10;
  // The cross track error is calculated by evaluating at polynomial at x, f(x)
  // and subtracting y.
  double cte = polyeval(coeffs, x) - y;
  // Due to the sign starting at 0, the orientation error is -f'(x).
  // derivative of coeffs[0] + coeffs[1] * x -> coeffs[1]
  double epsi = psi - atan(coeffs[1]);
  //**计算非线性规划求解时间**
  clock_t start, finish;
  double duration;

  Eigen::VectorXd state(6);
  state << x, y, psi, v, cte, epsi;

  std::vector<double> x_vals = {state[0]};
  std::vector<double> y_vals = {state[1]};
  std::vector<double> psi_vals = {state[2]};
  std::vector<double> v_vals = {state[3]};
  std::vector<double> cte_vals = {state[4]};
  std::vector<double> epsi_vals = {state[5]};
  std::vector<double> delta_vals = {};
  std::vector<double> a_vals = {};

  for (size_t i = 0; i < iters; i++) {
    std::cout << "Iteration " << i << std::endl;
    start = clock();
    auto vars = mpc.Solve(state, coeffs);
    finish = clock();
    duration = (double)(finish - start) / CLOCKS_PER_SEC;

    std::cout << "TIME = " << duration << std::endl;
//**保存预测步长范围状态及输入，用于绘图**
    x_vals.push_back(vars[0]);
    y_vals.push_back(vars[1]);
    psi_vals.push_back(vars[2]);
    v_vals.push_back(vars[3]);
    cte_vals.push_back(vars[4]);
    epsi_vals.push_back(vars[5]);

    delta_vals.push_back(vars[6]);
    a_vals.push_back(vars[7]);

    state << vars[0], vars[1], vars[2], vars[3], vars[4], vars[5];
    std::cout << "x = " << vars[0] << std::endl;
    std::cout << "y = " << vars[1] << std::endl;
    std::cout << "psi = " << vars[2] << std::endl;
    std::cout << "v = " << vars[3] << std::endl;
    std::cout << "cte = " << vars[4] << std::endl;
    std::cout << "epsi = " << vars[5] << std::endl;
    std::cout << "delta = " << vars[6] << std::endl;
    std::cout << "a = " << vars[7] << std::endl;
    std::cout << std::endl;
  }

  // Plot values
  // NOTE: feel free to play around with this.
  // It's useful for debugging!
  plt::subplot(3, 1, 1);
  plt::title("CTE");
  plt::plot(cte_vals);
  plt::subplot(3, 1, 2);
  plt::title("Delta (Radians)");
  plt::plot(delta_vals);
  plt::subplot(3, 1, 3);
  plt::title("Velocity");
  plt::plot(v_vals);

  plt::show();
}